The centre of the conic represented by the equation $2x^2 - 72xy + 23y^2 - 4x - 28y - 48 = 0$ is

The equation $\frac{x^2}{1-k} - \frac{y^2}{1+k} = 1$,where $k > 1$,represents:

In order to eliminate the first degree terms from the equation $2x^2+4xy+5y^2-4x-22y+7=0$,the point to which the origin is to be shifted is:

What type of conic section is represented by the equation $13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0$?

Consider the lines $L_1$ and $L_2$ defined by $L_1: x \sqrt{2} + y - 1 = 0$ and $L_2: x \sqrt{2} - y + 1 = 0$. For a fixed constant $\lambda$,let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y = 2x + 1$ meets $C$ at two points $R$ and $S$,where the distance between $R$ and $S$ is $\sqrt{270}$. Let the perpendicular bisector of $RS$ meet $C$ at two distinct points $R^{\prime}$ and $S^{\prime}$. Let $D$ be the square of the distance between $R^{\prime}$ and $S^{\prime}$.
$(1)$ The value of $\lambda^2$ is
$(2)$ The value of $D$ is

To convert the equation $2x^2 + 4xy + 5y^2 - 4x - 22y + 29 = 0$ to homogeneous form,the origin is shifted to the point: